CMGT130
MATHEMATICS AND STATISTICS APPRECIATION
Section 2 of the Course Notes
Carolyn Kennett
Learning Centre for Numeracy Skills
Macquarie University
1
Contents
1
Fractions
3
2
Percentages
7
3
Interchanging Fractions, Decimals and Percentages
9
4
Personal Diary Entry
12
5
More Problem Solving
13
6
Introductory Algebra
15
7
Further Algebra
18
8
Personal Diary Entry
21
ACKNOWLEDGEMENTS
I wish to thank Liz Spielman for the original development of this course and
Pamela Shaw for much of the material contained in these workbooks. The personal
diary sheets have been based on a tertiry preparation program at the University of
Southern Queensland. Data for some of the statistical exercises has been provided by
former students. Also, without funding from Macquarie University the development
of this course would not have been possible.
Carolyn Kennett, 2000
2
1
Fractions
For the rst activity we will do you should work in twos or threes. You will need a
number of strips of paper and a pair of scissors. These are all to be folded and then
cut as shown below.
Mark the rst strip as shown:
1 whole
Fold the second strip into two equal parts and cut along the fold line. Label each
strip.
1
2
1
2
Repeat this halving process until you have the following strips laid out on your
table.
1 whole
1
2
1
2
1
4
1
4
1
8
1
16
1
8
1
16
1
16
1
4
1
8
1
16
1
16
1
8
1
16
1
16
1
4
1
8
1
16
1
16
1
8
1
16
1
16
1
8
1
16
1
16
1
8
1
16
1
16
1
16
Notice how we have been writing the fractions. The denominator or bottom
part of the fraction is the number of equal parts that we have divided our whole into.
Thus, when we divided our whole piece into eight equal parts and considered one
piece the denominator was eight and the piece was 18 of the whole.
Three eighths ( 38 ) would be three (3) of the one eighth ( 18 ) pieces.
Take three of the one quarter( 14 ) pieces and lay them on top of the whole strip.
You have now covered three quarters of the whole strip. We write this as 34 . The 3
or the top part of the fraction is referred to as the numerator. 43 means that we
have divided the whole into four equal parts and taken three of them.
3
Exercise
Find ten other combinations of the strips in front of you which could be used to cover
three quarters of the strip. Record each combination as a sum to 43 .
In the gures below, what fraction is the shaded area of the whole gure?
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Shade 32 (two thirds) of the gure below. Compare your shading with your neighbour's.
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You should have found from the exercise above that 23 was equivalent to 128 . You
may also have found a number of equivalent ways of expressing three quarters. You
4
probably found that six of the 18 pieces and twelve of the
same length as 34 . We could write this as
1
16
pieces were exactly the
3 6 12
= =
4 8 16
These are referred to as equivalent fractions. These fractions can be found by
multiplying 34 by dierent representations of 1. i.e. by multiplying by 1 in disguise.
e.g. 22 = 1. Hence
3 32
6
=
=
4 42
8
12
3 34
=
=
4 44
16
and
Exercises
1. Can you nd two equivalent fractions for each of the following?
1
3
2
=
=
=
=
=
=
5
8
3
Compare your answers with the rest of your group. Try entering each of your
answers into the calculator. Press =. What do you notice?
2. Use your calculator to simplify the following fractions
3
9
=
3
25
100
=
5
21
28
=
15
25
=
5
10
40
=
1
36
24
= 12
4
3. Complete the following without your calculator
1
4
=
8
2
5
=
20
3
5
=
25
1
3
=
30
1
2
=
6
3
4
=
12
4
=
21
28
2
=
36
24
4. Complete the following
6
9
=
3
25
100
=
5
4
=
10
40
15
25
=
3
5. In a group of 60 students, 27 were mature age students. What fraction of the
group were mature age students? Can you re-express the fraction that you
obtained with a smaller denominator?
5
Addition of Fractions
Back to the strips. Take one 18 and put it along side another 18 . Now nd one single
piece that is equal to these two pieces.
You will have found out that
1 1 1
+ =
8 8 4
Of course it is also equal to
2
8
Try the following:
1
16
+ 161 =
1
2
+ 14 =
1
16
+ 18 + 161 =
+ 161 =
1
4
+ 18 + 161 =
1
4
+ 18 =
1
4
Comparing Fractions
The sign
example:
>
means ' is greater than', while the sign
4<8
and
<
means 'is less than'. For
5>2
Using your calculator, convert each of the following fractions to decimals and decide
which fraction is larger. In the gap between the fractions, put in the correct sign.
Fractions
Decimals
Fractions
3
4
2
5
7
8
4
5
1
4
1
5
5
9
6
10
Decimals
Multiplication of Fractions
1. What fraction of the whole will you have if you have 5 of the
1
16
2. What fraction of the whole will you have if you have 5 of the
1
8
3. What if you had 5 of the
1
4
pieces?
pieces?
pieces?
4. How long will the resulting piece be if you fold the
1
16
piece in half?
5. Express all of the above as multiplication of fractions.
6. Make up two examples of multiplication of fractions and give them to your
neighbour to solve. Solve theirs.
6
2
Percentages
Percentages are fractions with a denominator of 100.
In the diagram below, what fraction of the total number of small squares have
black blobs on them? i.e. what percentage of the squares have black blobs in them?
yyyy
yyyy
y
y
y
y
y y y y
y
y
y
y
y
y
Often there will not be 100 squares or 100 people out of which to express a fraction
or a percentage. When this is the case you will need to nd an equivalent fraction
out of 100 by multiplying by 100% or by multiplying by 1.
In the next gures, what percentage of the squares have circular shapes drawn in
them?
~
~ ~
~ ~
~ ~
~
~
~ ~ ~
~ ~
Percentage circles =
~
Percentage circles =
As we have said previously a percentage is just a fraction with a denominator of
100, thus,
1
15
5
=
=
= 5%
20 20 5 100
3
3 10
30
=
=
= 30%
10 10 10 100
2 2 20
40
=
=
= 40%
5 5 20 100
1
1
100
= 100% =
% = 5%
20 20
20
3
3
300
= 100% =
% = 30%
10 10
10
2 2
200
= 100% =
% = 40%
5 5
5
or
or
or
Therefore you can either nd out how many times the denominator goes into 100
and multiply the top and bottom of the fraction by this number or multiply the
numerator by 100, then divide through by the denominator.
7
For this activity you will need a metre ruler, coloured strips representing halves,
thirds, quarters, fths, tenths, twentieths and twenty-fths with each whole making up one metre in length. Use the strips to convert the following fractions to
percentages:
3
4
=
7
8
=
1
5
=
13
20
=
4
25
=
3
5
=
1
4
=
17
20
=
7
25
=
Use the strips to convert each percentage to a fraction in its simplest form:
80% =
40% =
35% =
75% =
85% =
48% =
37:5% =
16% =
62:5% =
Now try to do the previous exercise mathematically.
8
3
Interchanging Fractions, Decimals and Percentages
Changing Fractions to Decimals
Using your calculator divide the top number (numerator) by the bottom number
(denominator) to express a fraction as a decimal. If you do not have a calculator we
can do it manually. Be sure to put a decimal point after the numerator and add a
few zeros.
3
can be expressed as
3:00 4 = 0:75
4
Changing Fractions to Percentages
Express the fraction as a decimal (as shown above) then multiply the result by 100%.
This can be done easily by shifting the decimal point two places to the right.
3
= 3:000 8 = 0:375 = 37:5%
8
When the denominator of the fraction to be converted goes evenly into 100, the
percentage can be found using equivalent fractions.
3
=
5 100
60
3 20
=
= 60%
5 20 100
Converting Percentages to Decimals
In order to convert a percentage into a decimal, divide be 100. This can be done
easily by shifting the decimal point two places to the left. If there is no decimal point
be sure to place a decimal point to the right of the whole number.
45:8% = 0:458
and
0:5% = 0:005
and
7% = 7:0% = 0:07
Ordering Fractions, Decimals and Percentages
In order to compare numbers they must all be presented in the same form with
the same number of decimal places. It is usually easiest to convert everything to
decimals. Once you have done this, write the numbers underneath each other lining
up the decimal points. Fill in any blanks with zeros. Compare the whole number
side rst. If there is a match, then compare the fractional side. For example: Express
the following numbers in order from smallest to largest.
0:5; 5:3%; 0:54; 47%;
9
3 46
;
; 3
5 100
1. Convert all the numbers to decimals:
0:5; 0:053; 0:54; 0:47; 0:6; 0:46; 3
2. Line the numbers up
0:5
0:053
0:54
0:47
0:6
0:46
3
3. Fill in all of the blanks with zeros
0:500
0:053
0:540
0:470
0:600
0:460
3:000
4. Compare the whole numbers. If there is a match, compare the fractional side.
Since six of the numbers begin with a 0: we must compare the right hand side.
Clearly
53 < 460 < 470 < 500 < 540 < 600
5. Express the numbers from smallest to largest
3
46
; 47%; 0:5; 0:54;
; 3
5:3%;
100
5
Changing Mixed Numbers to Improper Fractions
3 45 can be expressed as 195 since there are 15 fths in 3 wholes plus 4 fths. This can
be worked out by multiplying the whole number by the denominator and adding the
numerator.
Changing Improper Fractions to Mixed Numbers
Improper fractions are fractions in which the numerator is larger than the denominator. For example 125 . You can use the fraction key on your calculator to convert
these to mixed numbers or you can divide by hand. For example:
12 5 = 2 remainder 2
The remainder can be expressed as the fraction 25 . So
10
12
5
= 2 52 .
Multiplying Fractions
Before performing any operations on fractions always represent it in the form ab even
if this makes it an improper fraction. Once all fractions are in this form simply
multiply the numerators together and then multiply the denominators together. For
example:
3 1 18 18 9
4
One half of 3 = =
= =1
5 2 5
10 5
5
Finding a Percentage
When you are asked to nd a percentage you must nd the fraction rst. For instance
if there are 15 men, 13 women and 22 children in a group, the fraction of men in the
group is 15
50 since there are 50 = 15 + 13 + 22 people in the group altogether. You can
either multiply by 100 % (if you have a calculator) or nd the equivalent fraction
out of 100. In both cases you will reach an answer of 30%.
Ratio
15
can be written as 15:50. However
Any fraction can also be expressed as a ratio. 50
be careful of the wording of these types of questions. The ratio of men to the total
number in the group is 15:50 but the ratio of men to women is 15:13.
11
4
Personal Diary Entry
1. What is the most important thing you have learnt from these topics?
2. Write down one thing you still nd diÆcult.
3. At the moment how are you feeling about studying maths? Circle any appropriate answers and/or add your own.
interested
relaxed
worried
successful
confused
clever
happy
bored
rushed
4. How could you improve your understanding of the maths covered in these
topics?
5. Comment generally about the course so far and how you are progressing.
Thank you
12
5
More Problem Solving
Below are a number of problems. They are designed to give you practice at solving
problems and in asking questions and making suggestions that will help your partner
solve problems.
Take turns at being the problem solver and the helper. Try to do at least two
problems each.
Responsibilities of the Problem Solver
You may select any of the problems below to solve. Read the problem to your
partner as many times as you need to fully understand it. Start solving the problem
and tell your partner everything that you are thinking of as you solve the problem.
Don't expect to be able to solve the problem straight away. When you have nished
check that your answer satises the question.
Responsibilities of the Listener
Listen carefully to the problem solver as she both reads the problem and tells you
what she is thinking. If you do not understand what she is saying ask for clarication.
(Eg What do you mean? Can you explain that in another way? Would you repeat
that? Could you go through that again more slowly?)
Encourage the problem solver to describe what she is thinking.
(Eg What are you thinking? What are you writing down? Can you explain your
diagram?)
Check that your partner is being accurate.
(Eg Are you sure that that is correct? I can't follow that.)
DO NOT tell the problem solver how to do the problem.
DO NOT tell the problem solver how to correct an error.
DO NOT give hints.
DO be encouraging.
13
Problem Set
1. Bob owns 6 suits, 3 less than James and twice as many as Peter. Graham owns
3 times as many suits as James. How many suits each do Graham and Peter
own?
2. On a certain day I ate lunch at Marxines, took out 2 books from the library,
visited the museum and had a cavity lled. Marxines is closed on Wednesday, the library is closed on the week-end, the museum is open only Monday,
Wednesday and Friday and my dentist has oÆce hours, Tuesday, Friday and
Saturday. On which day of the week did I do all of these things?
3. Continue this pattern 2 7 10 15 18 23 26 31 34 39 ......
4. Continue this pattern 9 12 11 14 13 16 15 18 ......
5. A certain ball when dropped from any height, bounces one third of the original
height. If the ball is dropped from 54 feet, bounces back up and continues
to bounce up and down, what is the total distance that the ball has travelled
when it hits the ground for the fourth time?
6. Paul, Nick and Tom dier in height. Their last names are Smith, Jones and
Chan but not necessarily in that order. Paul is taller than Tom but shorter
than Nick. Smith is the tallest of the three and Cahn is the shortest. What
are Paul's and Tom's last names?
7. Two workers, working 9 hours made 243 parts. One of them made 13 parts
every hour. How many parts did the other one make in an hour?
8. What number is twice the distance below 20 as 7 is above 4?
9. A family of two uses 20 litres of water every week day. On Saturday this norm
is increased by 15% but on Sundays it is decreased by 7:5%. In how many
weeks will the family use 424:5 litres of water?
10. Continue this pattern 5 10 20 40 80 160 .....
14
6
Introductory Algebra
In Mathematics we are continually looking for patterns and for relationships. In
addition, we try to generalise and we can frequently do this with the help of algebra.
(This is more of the kind of shorthand and simplication that we mentioned in
connecton with powers of ten.)
Try the examples below.
1. What are the next two numbers in the sequence?
(a)
(b)
(c)
(d)
(e)
(f)
(g)
2,4,6,8,10,
10,20,30,40,
3,6,9,12,15,
10,102 ,103 ,104 ,
1,4,9,16,25,36,
1,2,4,7,11,16,22,
Make up a sequence of your own. Ask your neighbour if they can nd the
next two numbers in your sequence.
2. Use the matches to construct a string of three squares.
t
t
t
t
t
tt
t
t
t
tt
How many matches did you use? Can you nd some methodical way of counting
the number of matches used? How many matches would you have used for
(a)
(b)
(c)
(d)
5 squares
8 squares
16 squares
In words explain how the number of matches is related to the number of
squares.
How did your neighbour explain this? Did she use the same method as you?
If she used a dierent method do you understand it? Find somebody else who
used another method again. Hopefully you all arrived at the same answer even
though you used dierent methods.
15
3. A Magic Square consists of numbers arranged in the form of a square so that
the sum of the numbers in each row, in each column and along each diagonal
is the same. Magic squares have been in existence for some thousands of years
and were used as magic charms in Europe in the middle ages. The oldest magic
square known is shown below.
4
9
2
3
5
7
8
1
6
Check that it is a magic square.
Now look at the square below, it has algebraic expressions in each of the smaller
squares. By making the substitions ll in the lower square. ( When we make
a substitution we put in particular numbers in place of each of the letters. For
example, substituting x = 5 and y = 3 into the expression x + y gives 5 + 3
which equals 8.)
x
x
z
x
+y+z
x
y
+z
y
x
x
+y
x
x
z
+y
y
x
z
+z
Substitute x = 5, y = 3, z = 1 into the algebraic expressions above and put
your results in the corresponding squares below.
What do you notice?
Try another set of values of x, y and z with x > (y + z ). What would happen
if x < (y + z )? Try a set of values.
16
4. For each of the gures below
(a) Count the number of vertices or corners (ie points)
(b) Count the number of edges (knife edges)
(c) Count the number of faces (at surfaces)
and use this information to complete the table below. Remember that there
are vertices, edges and faces that you can't see.
SBB S
BS
BB S
B
Tetrahedron
@@
@
Triangular Prism
eB
BBee
B e
BB ee
B BB
Square Pyramid
Name of Figure
Cube
@
@
@
Hexagonal Prism
Number Number Number
of vertices of edges of faces
v
e
f
+v
e
f
If you have counted correctly you should have found that f + v e is always
equal to 2. You will therefore have veried Euler's Theorem which states
that for any solid polyhedron if we denote the number of faces by f , the number
of vertices or corners by v and the number of edges by e. Then f + v
e = 2.
17
7
Further Algebra
1. Take three cuisenaire rods all of the same colour. Let y be the length in cm
of each rod. Put the rods in a line touching end to end. Use a ruler marked
in centimetres to measure the length of the three rods. For your three rods, 3
lots of y equals what? You can express this as 3y =
In arithmetic we write 8 4. In algebra we could write 8 a where a is a
variable but using our shorthand we instead often write 8a. Instead of 1a we
write a. Instead of 7 a we would write what?
Compare your result for 3y with other members of your group. Probably they
will have got dierent results to yourself. Why? We refer to y , which can take
dierent values, as a variable.
2. Still using the same coloured rods make a model of 2y . How long is it? Could
you have worked this out without measuring it?
By using the small centicubes model 2y +3. Use a ruler measured in centimetres
to nd what this equals? What value did another member of your group get
for this?
3. Model and draw 4y + 2. What does it equal? Rearrange your model so that
you have two equal halves. The value of course stays the same.
Try to model and draw 2 lots of 2y + 1. What does this equal? The mathematical way of writing this is 2(2y + 1). Compare this model with the one of
4y + 2.
Recall that in Block 3 we talked about the distributive law. Can you see the
similarity betweeen what we did then and what we are doing now?
4. Complete the table below using dierent values for y . i.e. substitute y equals
dierent values in each row of the table. The rst one has been done for you.
I had the black rod so my y equals 7.
y
7
y
+5
12
y
+ 1 2(y + 1) 2y + 2 3y + 6 3(y + 2)
8
16
16
27
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In the formula for calculating compound interest
I
= P (1 + r)t
, P , r and t are variables. Where I represents the amount of money you
have after getting your interest, P is the amount of money you started with,
i.e. the principal, r is the interest rate written as a decimal rather than a
I
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percentage and t is the number of time intervals that we are calculating over.
So for example if I put $1000 in an account that paid 6% annually and I left
the money there for 3 years, then P = 1000, r = 0:06 and t = 3. I would be
how much I would have in the account at the end of three years. We say that
I is a function of t when the principal and interest are known.
5.
+ 5 is an algebraic expression and x is the variable. We can draw up a table
of x-values and their corresponding x + 5 values.
x
x
x
+5
3
8
2
7
6
11
1
4
6
1
Think of some more values of x. Write them in the top line. Calculate the
corresponding values of x + 5 and write them in the corresponding place in the
table.
Similarly we can draw up a table relating y and 3y 2 + 2.
y
1
6
0
1
3y 2 + 2
3 12 + 2
3 36 + 2
30+2
31+2
= 3y 2 + 2
=5
= 110
=2
=5
Below we have a number of examples showing further the ability of algebra to
generalise.
6. If z is an integer (i.e. a whole number) and we add 3 to it the new integer will
be z + 3.
(a) What would the new integer be if we added 9 to it instead?
(b) What would the new integer be if we had added 20 instead?
(c) What would the new integer be if we had instead multiplied by 5?
7. In the following examples write an algebraic expression for each statement.
The rst is done for you.
(a) Let p be a number, multiply it by 6 and then add 4 to the result.
6p + 4
(b)
(c)
(d)
(e)
Let x be a number, then add 5
Let f be a number, then subtract 10.
Let q be a number, square it and then subtract 3.
Let y be a number, square it, multiply the result by 5 and then add 12.
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8. Think of a number between one and ten. Double it. Add 8. Multiply by 3.
Subtract 12. Divide by 6. Subtract the number you rst thought of. The
answer is?
Try to model it with blocks. Now try it with algebra. What did you notice?
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8
Personal Diary Entry
1. What is the most important thing you have learnt from these topics?
2. Write down one thing you still nd diÆcult.
3. At the moment how are you feeling about studying maths? Circle any appropriate answers and/or add your own.
interested
relaxed
worried
successful
confused
clever
happy
bored
rushed
4. How could you improve your understanding of the maths covered in these
topics?
5. Comment generally about the course so far and how you are progressing.
Thank you
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